Critical temperature shift modeling of confined fluids using pore-size-dependent energy parameter of potential function

The behavior and critical properties of fluids confined in nanoscale porous media differ from those of bulk fluids. This is well known as critical shift phenomenon or pore proximity effect among researchers. Fundamentals of critical shift modeling commenced with developing equations of state (EOS) based on the Lennard–Jones (L–J) potential function. Although these methods have provided somewhat passable predictions of pore critical properties, none represented a breakthrough in basic modeling. In this study, a cubic EOS is derived in the presence of adsorption for Kihara fluids, whose attractive term is a function of temperature. Accordingly, the critical temperature shift is modeled, and a new adjustment method is established in which, despite previous works, the bulk critical conditions of fluids are reliably met with a thermodynamic basis and not based on simplistic manipulations. Then, based on the fact that the macroscopic and microscopic theories of corresponding states are related, an innovative idea is developed in which the energy parameter of the potential function varies with regard to changes in pore size, and is not taken as a constant. Based on 94 available data points of critical shift reports, it is observed that despite L–J, the Kihara potential has sufficient flexibility to properly fit the variable energy parameters, and provide valid predictions of phase behavior and critical properties of fluids. Finally, the application of the proposed model is examined by predicting the vapor–liquid equilibrium properties of a ternary system that reduced the error of the L–J model by more than 6%.


S1.2
Here we aim to explain why and critical temperature are in direct relation to each other, and why emulates the behavior of when it is decreased because of the confinement. Since John. M. Prausnitz has explained this relation perfectly [1], we bring their discussions on "Molecular Theory of Corresponding States" where a relation is established between the microscopic and microscopic theories of corresponding states using the concept of Canonical Partition Function:

Molecular Theory of Corresponding States
Classical or macroscopic theory of corresponding states was derived by van der Waals based on his well-known equation of state. It can be shown, however, that van der Waals' derivation is not tied to a particular equation but can be applied to any equation of state containing two arbitrary constants in addition to gas constant R.
From the principle of continuity of the gaseous and liquid phases, van der Waals showed that at the critical point These relations led van der Waals to the general result that for variables  (volume), T (temperature), and P (pressure) there exists a universal function such that ( . , ) = 0 (4-63) S1.3 is valid for all substances; subscript c refers to the critical point. Another way of stating this result is to say that, if the equation of state for any one fluid is written in reduced coordinates (i.e., /C, T/TC, P/Pc), that equation is also valid for any other fluid.
Classical theory of corresponding states is based on mathematical properties of the macroscopic equation of state. Molecular or microscopic theory of corresponding states, however, is based on mathematical properties of the potential-energy function.
Intermolecular forces of a number of substances are closely approximated by the inverse-power potential function given by Eq. (4-24). The independent variable in this potential function is the distance between molecules. When this variable is made dimensionless, the potential function can be rewritten in a general way such that the dimensionless potential is a universal function F of the dimensionless distance of separation between molecules: where i is an energy parameter and i is a distance parameter characteristic of the interaction between two molecules of species i. For example, if function F function F is given by the Lennard-Jones potential, then i is the energy (times minus one) at the potential-energy minimum, and I is the distance corresponding to zero potential energy.
However, Eq. (4-64) is not restricted to the Lennard-Jones potential, nor is it restricted to an inverse-power function as given by Eq. (4-24). Equation (4-64) merely states that the reduced potential energy (ii/i) is some universal function of the reduced distance (r/i). S1.4 Once the potential-energy function of a substance is known, it is possible, at least in principle, to compute the macroscopic configurational properties of that substance by the techniques of statistical mechanics. Hence a universal potential-energy function, Eq.
(4-64), leads to a universal equation of state and to universal values for all reduced configurational properties.
To obtain macroscopic thermodynamic properties from statistical mechanics, it is useful to calculate the canonical partition function of a system depending on temperature, volume, and number of molecules. For fluids containing small molecules, the partition function Q is expressed as a product of two factors, where the translational contributions to the energy of the system are separated from all others, due to other degrees of freedom such as rotation and vibration. It is assumed that contributions from rotation and vibration depend only on temperature. These contributions are called internal because (by assumption) they are independent of the presence of other near-by molecules.
In the classical approximation, the translational partition function, Qtrans, splits into a product of two factors, one arising from the kinetic energy and the other from the potential energy. For a one-component system of N molecules, Qtrans is given by where m is the molecular mass, k is Boltzmann's constant, h is Planck's constant, and t (r1, …, rN) is the potential energy of the entire system of N molecules whose positions are described by vectors r1, …, rN. For a given number of molecules and known molecular mass, the first factor depends only on the temperature. The second factor, called the configurational integral, ZN, depends on temperature and volume: Hence the configurational part provides the only contribution that depends on intermolecular forces. However, ZN is not unity for an ideal gas (t=0). For an ideal gas, = .
The where NA is Avogadro's constant (c is per mole) and c1, c2, and c3 are universal constants.
For simple nonpolar molecules, i.e., those nonpolar molecules having a small number of atoms per molecule, these relations have been found empirically for the case where the generalized function F is replaced by the Lennard-Jones (12-6) potential (Hirschfelder et al., 1964). For that particular case, we have, approximately, Similarly, the critical volume reflects the size of the molecules; hence, the proportionality between distance parameter  3 and the critical volume is also reasonable. The proportionality of the critical pressure to the ratio / 3 follows because, according to the theory, the compressibility factor zc is the same for all fluids.
You can see that, the critical point is known as the characteristic state, where the liquid and gaseous phases of a substance become identical. At this point, the critical temperature is a measure of kinetic energy, therefore a relation between and is quite expectable and reasonable. From the foregoing explanations, knowing that and are of one type, and based on the shift of critical temperature of confined fluids, we can infer that the energy parameter of the potential function would accordingly shift. In other words, as a component S1.10 has different critical temperature values at different pore radii, there might be also a possibility for to vary in pore size which is designated as herein.
So why manipulating rather than or ? Along the lines of and , there is a relation between the size parameters or and the critical molar volume ( ), because the latter represents the molecular size. The point is, however, that of a bulk fluid does not really differ from the confined one. Not only this claim is theoretically conspicuous, but also experiments have made the same assertion as well. Even if there were any shift of regarding pore shrinking, it would be irrelevant to choose a distance parameter for calibrating an energy parameter ( ). Thus, (or in fact ) seems to be the only choice for manipulation. References: